Coalgebraic Infinite Traces and Kleisli Simulations

Kleisli simulation is a categorical notion introduced by Hasuo to verify finite trace inclusion. They allow us to give definitions of forward and backward simulation for various types of systems. A generic categorical theory behind Kleisli simulation has been developed and it guarantees the soundness of those simulations with respect to finite trace semantics. Moreover, those simulations can be aided by forward partial execution (FPE)---a categorical transformation of systems previously introduced by the authors. In this paper, we give Kleisli simulation a theoretical foundation that assures its soundness also with respect to infinitary traces. There, following Jacobs' work, infinitary trace semantics is characterized as the "largest homomorphism." It turns out that soundness of forward simulations is rather straightforward; that of backward simulation holds too, although it requires certain additional conditions and its proof is more involved. We also show that FPE can be successfully employed in the infinitary trace setting to enhance the applicability of Kleisli simulations as witnesses of trace inclusion. Our framework is parameterized in the monad for branching as well as in the functor for linear-time behaviors; for the former we mainly use the powerset monad (for nondeterminism), the sub-Giry monad (for probability), and the lift monad (for exception).Comment: 39 pages, 1 figur

A Boyer-Moore Type Algorithm for Timed Pattern Matching

The timed pattern matching problem is formulated by Ulus et al. and has been actively studied since, with its evident application in monitoring real-time systems. The problem takes as input a timed word/signal and a timed pattern (specified either by a timed regular expression or by a timed automaton); and it returns the set of those intervals for which the given timed word, when restricted to the interval, matches the given pattern. We contribute a Boyer-Moore type optimization in timed pattern matching, relying on the classic Boyer-Moore string matching algorithm and its extension to (untimed) pattern matching by Watson and Watson. We assess its effect through experiments; for some problem instances our Boyer-Moore type optimization achieves speed-up by two times, indicating its potential in real-world monitoring tasks where data sets tend to be massive

Fair Simulation for Nondeterministic and Probabilistic Buechi Automata: a Coalgebraic Perspective

Notions of simulation, among other uses, provide a computationally tractable and sound (but not necessarily complete) proof method for language inclusion. They have been comprehensively studied by Lynch and Vaandrager for nondeterministic and timed systems; for B\"{u}chi automata the notion of fair simulation has been introduced by Henzinger, Kupferman and Rajamani. We contribute to a generalization of fair simulation in two different directions: one for nondeterministic tree automata previously studied by Bomhard; and the other for probabilistic word automata with finite state spaces, both under the B\"{u}chi acceptance condition. The former nondeterministic definition is formulated in terms of systems of fixed-point equations, hence is readily translated to parity games and is then amenable to Jurdzi\'{n}ski's algorithm; the latter probabilistic definition bears a strong ranking-function flavor. These two different-looking definitions are derived from one source, namely our coalgebraic modeling of B\"{u}chi automata. Based on these coalgebraic observations, we also prove their soundness: a simulation indeed witnesses language inclusion